Tonelli Hamiltonians without conjugate points and integrability

We prove that all the Tonelli Hamiltonians defined on the cotangent bundle of the -dimensional torus that have no conjugate points are integrable, i.e. is foliated by a family of invariant Lagrangian graphs. Assuming that the Hamiltonian is , we prove that there exists a subset of such that the dynamics restricted to every element of is strictly ergodic. Moreover, we prove that the Lyapunov exponents of every integrable Tonelli Hamiltonian are zero and deduce that the metric and topological entropies vanish.